This will enter "Mark Mode" and allow you to select lines in the terminal screen.

To Paste:

This is pretty straight forward:

Click and drag your selection

Right Click to copy

Right Click again to paste (in the terminal), or just ctrl + v in another app to paste from clipboard

What you're looking for is bullet number 2. Once selected, right clicking will save this to your Windows clipboard. Right clicking in the cygwin terminal again will paste (if no selection is currently highlighted).